Question

. The vectors a and b are given by a=2pi-6pj+3pk and b=i+j-2k, where p is a positive constant. Given that a is a unit vector, find the exact value of p, (ii) find the value of la.b| and give a geometrical interpretation of |a.b|​

Answer 1

Explanation:

(i) To find the value of p, we need to find the magnitude of vector a:

|a| = √(2p^2i - 6p^2j + 3p^2k)

= √(2p^2 + 36p^2 + 9p^2)

= √(47p^2)

Since a is a unit vector, the magnitude of a must be 1. So, we can set up an equation to solve for p:

1 = √(47p^2)

p^2 = 1 / 47

p = √(1 / 47)

(ii) To find the dot product of a and b, we can use the formula:

a.b = |a| * |b| * cos(θ)

where θ is the angle between vectors a and b.

Since we know that a is a unit vector, the magnitude of a is 1. We can find the magnitude of b using the formula:

|b| = √(i^2 + j^2 + (-2k)^2)

= √(1 + 1 + 4)

= √(6)

So,

a.b = 1 * √(6) * cos(θ)